Innovative use of technology in the teaching of calculus

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use of technology in the teaching of maxima and minima

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Innovative use of technology in the teaching of calculus

  1. 1. 0011 0010 1010 1101 0001 0100 1011 1 2 4 SUBTOPIC: APPLICATIONS OF MAXIMA AND MINIMA PRESENTED BY : HIMANI ASIJA P.G.T. MATHEMATICS DELHI PUBLIC SCHOOL, VASANT KUNJ
  2. 2. EFFECTIVENESS OF USE OF TECHNOLOGY0011 0010 1010 1101 0001 0100 1011 For the best results of use of technology in education, it is required to 1 2 Carefully integrate technology and the content Select some abstract topics in the subject (math) and use technology as a medium of instruction 4 Focus on professional development of teachers (both in service and pre service) Make an intelligent choice on the kind of software to be used
  3. 3. REFLECTIONS ON THE USE OF0011 0010 TECHNOLOGY FROM AROUND 1010 1101 0001 0100 1011 THE WORLD  Is highlighted in the “Principles and Standards of Mathematics”, U.S.A. 1 2  Ministry Of Education, France since 2004, for future teachers 4  Ministry Of Education, Singapore from 2006 for senior classes and from 2007 for primary classes.
  4. 4. Use of technology to make teaching of calculus innovative0011 0010 1010 1101 0001 0100 1011 Subtopic: Applications of Maxima and Minima Problems discussed: 1 2 1. Kepler’s problem. 2. To find the minimum length a ladder through a vertical 4 fence of a given height and at a given distance from a wall. 3. Extension of the above problem with a circular fence.
  5. 5. Problem 1: Kepler’s problem0011 0010 1010 1101 0001 0100 1011 Statement : In 1612, the great scientist Johannnes Kepler was about to remarry and he needed to buy wine for the celebration. The wine salesman 1 2 brought barrels of wine to Kepler’s house and explained how they were priced: a rod was inserted into the barrel and diagonally 4 through a small hole in the top. When the rod was removed, the length of the rod which was wet determined the price for the barrel.
  6. 6. 0011 0010 1010 1101 0001 0100 1011 R 1 2 4H L
  7. 7. A MATHEMATICAL SOLUTION OF KEPLER’S PROBLEM0011 0010 1010 1101 0001 0100 1011 Length of the rod (L) = fixed, since Kepler wanted to keep his bugdet fixed. 1 2 Now, volume of the cylinder is given by V= Π R2 H Also, L2 = R2 + H2 SO, V = Π (L2 - H2 ) H 4 On finding the first derivative and equating it to zero, V´(H) = Π (L2 - 3H2 ) V´(H) = 0 1 H= L (Omitting the negative sign as h is positive) 3 FIND V˝(H) AND OBSERVE IT TO BE NEGATIVE
  8. 8. Using G.S.P. finding the solution of the problem0011 0010 1010 1101 0001 0100 1011 1 2 4
  9. 9. Problem 2: Minimum height of ladder through a fence0011 0010 1010 1101 0001 0100 1011 Statement: To find the minimum length of a ladder passing through a fence of given height at a given distance from the wall. WA 1 2 4 L L (y) LADDER (L) F E N a C E b x
  10. 10. A MATHEMATICAL SOLUTION0011 0010 1010 1101 0001 0100 1011 Assuming a=3 and b=3. Using geometry, y 3 x x 3 9x2 1 2 l2 x2 ( x 3) 2 On differentiation w.r.t. x, and equating the derivative to zero, 4 we obtain x=6 and y=6 and l=6√2 On finding second differential, at x=6, we observe that it is positive and hence ensures a minimum length of the ladder.
  11. 11. Using G.S.P. finding the solution of the problem0011 0010 1010 1101 0001 0100 1011 1 2 4
  12. 12. Problem 3 : Extension of the above problem with a circular fence0011 0010 1010 1101 0001 0100 1011 Statement: To find the minimum length of a ladder passing through a circular fence. 1 2 4 W LADDER A L L y r=3 FENCE x
  13. 13. A MATHEMATICAL SOLUTION0011 0010 1010 1101 0001 0100 1011 Given circular fence of radius r B Minimize length of the ladder BE=L C Let OB=y and OE=x, then by geometry, A AB=BC=y-3 and DE=CE=x-3 1 2 O So, we have L = x+y-6 and L2 = x2 +y2 D E Giving us L= y 2 6 y 18 y 6 4 On finding dL/dy, and putting it =0, y=6 3 2 y= 6+3 2 Satisfies the given condn.(i.e. finding L , it comes out to be positive, thus ensuring a minima.)
  14. 14. Using G.S.P. finding the solution of the problem0011 0010 1010 1101 0001 0100 1011 1 2 4
  15. 15. CONCLUSION AND RECOMMENDATIONS Technology emphasizes 10110011 0010 1010 1101 0001 0100discovery and minimizes lecturing and rote learning. Teacher student interaction increases 1 2 To make the use of technology most effective, it is important To correctly choose the topics to be taught using technology 4 To train the present teachers to use the technological aids To make it an integral part of the pre service teacher training To change the assessment techniques That educators and programmers work together Last, but not the least, to make the teachers believe that technology is not their replacement, but an aid for teaching.

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